Bonus Game Question

[123123123]

With pure guessing, is the odds of winning Bonus Game 25 percent or 50 percent?

I know this might be a dumb question, but after trying to explain to a friend for a good five minutes during today's episode why I thought it was 50%, I just want to make sure I wasn't being an idiot.

[Teddy]

Only one SP can have the bonus attached to it, so the chances of winning are 25%. Nothing hurts more than when you get three right and still lose the game (when the SP with the bonus was the only one you missed).

[Casey]

Neither of these numbers sound right. 

If I go up to play the game and I just randomly guess higher or lower for each of the small prizes, it's conceivable that I could get them all wrong.  The probability of getting any one small prize right is 50% but you have to multiply that one probability by 4 to get the right answer.  I'm sure someone can check my math but wouldn't the random chance of winning this game be 1/16 ?

[sideshowPA]

50% - and here's why.

The most important premise here is that the Bonus is placed before the game.  It does not take on a random position during the gameplay.  If we are OK with assuming this, then only one box matters during the gameplay - the box with the pre-determined BONUS.  The other 3 windows are.....well, window dressing.

So, assuming random guessing, the chance of winning is 50-50.

[MSTieScott]

50%, because, as sideshow said, the only window that has any bearing on the outcome of the game is the window which already contains the bonus. Although it occurs to me that even if the bonus window was randomly assigned, it would still work out to a 50% chance of winning.

25% is what the chance of winning would be if I presented you with four windows and said, "you have one opportunity to tell me which window has the bonus." But that's a different game altogether.

(If I wanted to be a jerk, I could point out that the 50%-by-random-guessing figure is assuming that "higher" and "lower" have been correct an equal number of times for the entire history of the game. I don't know whether that's the case.)

[Casey]

50% - and here's why.

The most important premise here is that the Bonus is placed before the game.  It does not take on a random position during the gameplay.  If we are OK with assuming this, then only one box matters during the gameplay - the box with the pre-determined BONUS.  The other 3 windows are.....well, window dressing.

So, assuming random guessing, the chance of winning is 50-50.

Of course.  Makes sense. 

[shell_game]

You can get to the 50% a different and more complicated way, but one that it sometimes more easy to understand for some folks.  If you think of each small prize answer as being either right or wrong, then you do have 16 different possible outcomes (RRRR, RRRW, RRWR,.... WWWW).  But suppose the prize with the bonus attached to it is the third prize.  You'll see in the list of 16 possible outcomes that this happens exactly 8 times, or 50% of the time.

Same idea if you used a probability tree.